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Mat. Fiz. Anal. Geom., 2005, Volume 12, Number 1, Pages 103–106 (Mi jmag174)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Notes

The Haar system in $L_1$ is monotonically boundedly complete

Vladimir Kadets

Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine

Abstract: Answering a question posed by J. R. Holub we show that for the normalized Haar system $\{h_n\}$ in $L_1[0,1]$ whenever $\{a_n\}$ is a sequence of scalars with $|a_n|$ decreasing monotonically and with $\sup_N\|\sum_{n=1}^N a_n h_n\| < \infty$, then $ \sum_{n=1}^\infty a_n h_n$ converges in $L_1[0,1]$.

Key words and phrases: Haar system; martingale; monotonically boundedly complete basis.

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Bibliographic databases:
MSC: 46B15, 60G46
Received: 13.08.2004
Language:

Citation: Vladimir Kadets, “The Haar system in $L_1$ is monotonically boundedly complete”, Mat. Fiz. Anal. Geom., 12:1 (2005), 103–106

Citation in format AMSBIB
\Bibitem{Kad05}
\by Vladimir Kadets
\paper The Haar system in $L_1$ is monotonically boundedly complete
\jour Mat. Fiz. Anal. Geom.
\yr 2005
\vol 12
\issue 1
\pages 103--106
\mathnet{http://mi.mathnet.ru/jmag174}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2135427}
\zmath{https://zbmath.org/?q=an:1091.46008}


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    This publication is cited in the following articles:
    1. V. G. Mikayelyan, “On a Property of the Franklin System in $C[0,1]$ and $L^1[0,1]$”, Math. Notes, 107:2 (2020), 284–287  mathnet  crossref  crossref  isi
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